2004 Impala wont start?

I have been trying to figure this out since last October and have had no luck.

One day the car would not start since the battery was dead, not sure what drained it. I tried to jump the car and I have only been able to get about a 1/4 crank or the engine turns over about a 1/4 of the way and stops. Same result over and over until one day in december it actually started! I drove it, parked it and again it would not start.

I changed the starter with rebuilt and had the old and rebuilt tested, both are ok.

After the 1/4 crank, the power in the cab goes out and comes back when I turn the key to off and back to on.

I posted these videos to help gauge the problem.

https://www.youtube.com/watch?v=OXIZ9WNbIR0

https://www.youtube.com/watch?v=aqWOYA1u5TY

If I should post somewhere else, please direct me. :) Any help is appreciated, I would hate to give the car to a salvage yard as it only has 130k and drives like a champ if I can get it started.